© annie patton differentiation of inverse trigonometric function next slide
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© Annie Patton
Differentiation of Inverse Trigonometric Function
1sin
1sin 1tan
1tan
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Aim of Lesson
To establish what an inverse trigonometric function is, then differentiate the inverse of sin
and tan from first principles and use these formulas.
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Inverse Trigonometric Function
1, then sin2 2
If Sin
It is the inverse of sin.
1
1
cos is the inverse of cos and
tan is the inverse of tan .
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1for the differentiation of sinx
Special rulea
1sin
sin
xy
ax
ya
sin
cos
x a y
dxa y
dy
2 2
2 2
1 1
cos cos
1
(1 sin )
dy
dx a y a y
a y
a is a constant
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-1
2 2
xdsindy 1a= =
dx dx a -x
This formula is given in the tables.
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1Differentiation of y=sin 4x
1sin 4y x 1
2 2
(sin ) 1with
x=4x and a=1
xd
aComparingdx a x
1
2 2 2 2 2
u=4x
(sin ) 1 1 14
1 ( ) 1 (4 ) 1 16
Let
du d u
dx du u x x
1
2 2
sin 1 44
1 16 1 16
dy d u du
dx du dx x x
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1for the differentiation of tanx
Special rulea
a is a constant
-1 xy=tan
a
2 2
dy 1 1 = =
dx a sec y a (1 + tan y)
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2
xThen tan y=
ax=a tan y
dx=a sec y
dy
2 2 2 2 2
2
1 1
( )(1 )
dy a
a a xdx a xxa
aa
This formula is given in the tables.
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Differentiate of y=tan-14x
u=4x 4du
Letdx
1
2 2
tan
1 1
1 1 16
y u
dy
du u x
2 2
1 4.4
1 16 1 16
dy dy du
dx du dx x x
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1 y=tan , find the value of at x= 2.dy
Given xdx
Leaving Certificate Higher No 6b(i) Paper 1 2004
1
2
tan
1
1
y x
dy
dx x
2
x= 2
1 1 1
1 2 31 ( 2)
At
dy
dx
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1 1 y=sin 10 , find the value of at x= .
20
dyGiven x
dx
Leaving Certificate Higher No 7 b (1) Paper 1 2002
1sin 10
u=10x
10
y x
Let
du
dx
2 2
1 1010
1 100 1 100
dy dy du
dx du dx x x
1 10 10 20at x= ,
20 100 3 31
400 4
dy
dx
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1
2 2
sin
1 1
1 1 100
y u
dy
du u x
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1 1
' '
2f(x)=tan and g(x)=tan , for x>0.
2
f ( ) g ( ).
xLet
x
Find x and x
'2
2( )
4f x
x
1
22
22
22
u xx
dux
dx x
1 2g(x)=tan
x
1
2
2
tan
1 141 1
y u
dy
du ux
2
2 2 2 2
2
1 2 2 2.
4 4 41
dy dy du x
dx du dx x x x xx
'2
2( )
4g x
x
Leaving Certificate Higher No 6c (i) Paper 1 2007
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Homework. Differentiate the following:
6.
-1
-1
-1
-1
-1
-1
1. y= sin 5x
2. y= tan 7x
x3. y= sin
2x
4. y=2x sin2
x5. y= tan
1+xy= tan (cos x)
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Conclusion
1
2 2
sin 1=
xd
adx a x
1 1
for example when y=4x+1. Let u=4x+1 then
sin (4 1) sin= .
And
d x d u du
dx du dx
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Conclusion
1
2 2
tanx
d aadx a x
1 1
for example when y=4x+1. Let u=4x+1 then
tan (4 1) tan= .
And
d x d u du
dx du dx
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•Note the differentiation of cos-1 is not on the syllabus.